G.N. de Oliveira

Matrices with prescribed characteristic polynomial and principal blocks. II

Abstract In this paper we present some new sufficient conditions for the existence of a square matrix with prescribed characteristic polynomial and two complementary principal submatrices.

Equality of decomposable symmetrized tensors and ∗-matrix groups

Let G be a subgroup of the full symmetric group Sn, and χ a character of G. A ∗-matrix can be defined as an n × n matrix B which satisfies dGx(BX) = dGx(X) for every n × n matrix X. They form a multiplicative group, denoted S(G,X), which plays a fundamental role in the study of equality of two decomposable symmetrized tensors. The main result of this paper (Theorems 2.2, 2.3, and 2.4) is a complete description of the matrices in S(G,X). This description has many consequences that we present. There are also results on related questions.

Conditions for equality of decomposable symmetric tensors. II.

Abstract We derive consequences of a condition for the equality of two star products given by the second author. We also study another method for the same problem which consists of comparing the components, in an appropriate basis, of the star products involved.

Note On the equality of star products

We study the problem of finding conditions for two star products of n vectors to be equal when the vectors in each star product are not linearly independent, and the problem of finding conditions for one star product to be zero. We reprove results of M. Marcus and J. Chollet in a very simple way. We present also some open problems.

Pairs of matrices satisfying certain polynomial identities

Abstract The main result describes the pairs (A,B) of matrices which satisfy d(AXB) d(X), where d(·) is a Schur function. We also study the semigroup of matrices A satisfying d(AX)d(XA).

Vectors with a prescribed minimal polynomial

Let V be a finite dimensional vector space over the field Fand φ (x)∊F[x].Letx V → V be a linear operator. Let Sφ be the set consisting of the vectors whose minimal polynomial φ(x)together with the zero vector We give necessary and sufficieni condition for S φ to be a subspace.

Note on the characteristic roots of tournament matrices

Abstract We give bounds for the real parts of the characteristic roots of tournament matrices.

Linear preservers of Schur functions

Abstract Let Mn be the vector space of the n × n matrices over the field F . Let H be a subgroup of the symmetric group of degree n and x an F -valued character of H. If A = [aij] ϵ Mn, the Schur function of A is A is d H x (A) = ∑ σϵH X(σ) Π i a iσ(i) . A linear mapping T: Mn → Mn is called a Schur function preserver if d H x (T(X)) = d H x (X) for every X in Mn. We survey some very recent results on Schur function preservers with special emphasis on the case in which H is the full symmetric group and F is the complex field. In this case a complete description of the preservers is possible.